A proof of four color conjecture (4CC)

2021 ◽

Author(s):

hongbo ZHUANG

Keyword(s):

foundout a critical corresponding relationship between a δwheelof any maximal simple plane graph of n vertices(nMSPG)and its corresponding δcycleof a corresponding n-1MSPGof the nMSPG, and relied on theinductive way, while with the help of KempeChain (KC), this article gives a brand-newlogical proof for the famous and morethan 100 years old fourcolor conjecture (4CC).

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A sufficient condition for a plane graph with maximum degree 6 to be class 1

Discrete Applied Mathematics ◽

10.1016/j.dam.2012.08.011 ◽

2013 ◽

Vol 161 (1-2) ◽

pp. 307-310 ◽

Cited By ~ 4

Author(s):

Yingqian Wang ◽

Lingji Xu

Keyword(s):

Maximum Degree ◽

Plane Graph ◽

Sufficient Condition ◽

Class 1

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GENERA OF THE LINKS DERIVED FROM 2-CONNECTED PLANE GRAPHS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216512501295 ◽

2012 ◽

Vol 21 (14) ◽

pp. 1250129 ◽

Cited By ~ 3

Author(s):

SHUYA LIU ◽

HEPING ZHANG

Keyword(s):

Explicit Formula ◽

Large Family ◽

Plane Graph ◽

Plane Graphs ◽

Degree Sum ◽

Oriented Link

In this paper, we associate a plane graph G with an oriented link by replacing each vertex of G with a special oriented n-tangle diagram. It is shown that such an oriented link has the minimum genus over all orientations of its unoriented version if its associated plane graph G is 2-connected. As a result, the genera of a large family of unoriented links are determined by an explicit formula in terms of their component numbers and the degree sum of their associated plane graphs.

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Polynomial algorithms for open plane graph and subgraph isomorphisms

Theoretical Computer Science ◽

10.1016/j.tcs.2013.05.026 ◽

2013 ◽

Vol 498 ◽

pp. 76-99 ◽

Cited By ~ 4

Author(s):

Colin de la Higuera ◽

Jean-Christophe Janodet ◽

Émilie Samuel ◽

Guillaume Damiand ◽

Christine Solnon

Keyword(s):

Plane Graph ◽

Polynomial Algorithms ◽

Open Plane

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A New Algorithm for Embedding Plane Graphs at Fixed Vertex Locations

The Electronic Journal of Combinatorics ◽

10.37236/10106 ◽

2021 ◽

Vol 28 (4) ◽

Author(s):

Marcus Schaefer

Keyword(s):

Plane Graph ◽

Plane Graphs ◽

Classic Result

We show that a plane graph can be embedded with its vertices at arbitrarily assigned locations in the plane and at most $6n-1$ bends per edge. This improves and simplifies a classic result by Pach and Wenger. The proof extends to orthogonal drawings.

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An Image Normalization Algorithm for Corner-Pointed Plane Graph

Journal of Convergence Information Technology ◽

10.4156/jcit.vol8.issue8.122 ◽

2013 ◽

Vol 8 (8) ◽

pp. 1030-1037

Author(s):

Mingxin Li ◽

Xiongfei Li ◽

Jinfeng Zhang

Keyword(s):

Plane Graph ◽

Image Normalization

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THE VIRTUAL DIMENSIONS OF A STRAIGHT LINE EMBEDDING OF A PLANE GRAPH

Lecture Notes Series on Computing - Algorithmic Aspects of VLSI Layout ◽

10.1142/9789812794468_0013 ◽

1993 ◽

pp. 357-363

Author(s):

TOSHIHIKO TAKAHASHI ◽

YOJI KAJITANI

Keyword(s):

Plane Graph ◽

Straight Line

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FOLDING EQUILATERAL PLANE GRAPHS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195913600017 ◽

2013 ◽

Vol 23 (02) ◽

pp. 75-92 ◽

Cited By ~ 1

Author(s):

ZACHARY ABEL ◽

ERIK D. DEMAINE ◽

MARTIN L. DEMAINE ◽

SARAH EISENSTAT ◽

JAYSON LYNCH ◽

...

Keyword(s):

Plane Graph ◽

Analogous Problem ◽

Central Vertex ◽

Polyhedral Complex ◽

Plane Graphs ◽

Single Vertex ◽

Np Completeness ◽

Np Complete ◽

Polyhedral Manifold

We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, it is known that such reconfiguration is not always possible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Furthermore, we show that the equilateral constraint is necessary for this result, by proving that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete.

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